Treasury Secretary Jacob Lew headed to Capitol Hill to both give and get a public scolding with his appearance before the committee promising to be yet another public restatement of the administration's stance that Congress needs to reopen the government and lift the U.S. borrowing cap before Obama will negotiate over the nation's budget ills.

WASHINGTON — President Obama today rejected a proposal from politically besieged House Republican leaders to extend the nation’s borrowing authority for six weeks because it would not also reopen the government. Yet both parties saw it as the first break in Republicans’ brinkmanship and a step toward a fiscal truce.

Twenty Republicans, led by Speaker John A. Boehner, went to the White House at Mr. Obama’s invitation after a day of fine-tuning their offer to increase the Treasury Department’s authority to borrow money to pay existing obligations through Nov. 22. In exchange, they sought the president’s commitment to negotiate a deal for long-term deficit reduction and a tax overhaul.

Mr. Boehner and his colleagues left after about an hour and a half without speaking to waiting reporters.

The Republican proposal could come to a vote as soon as Friday. But the White House and Congressional Democrats remained skeptical that House Republican leaders could pass the proposal. A large faction of Tea Party conservatives campaigned on promises never to vote to increase the nation’s debt limit, and say they do not believe the warnings — including from Republican business allies — that failing to act could provoke a default and economic chaos globally. And House Democrats vowed not to support the proposal without a companion measure to fully fund a government now shuttered for 10 days.

Arriving back at the Capitol, the House Republicans huddled in Mr. Boehner’s office for further discussion.

“We had a very useful meeting, and we expect further conversations tonight,” said Representative Eric Cantor of Virginia, the majority leader.

Despite the president’s rejection, the House Republican offer represented a significant breakthrough. Representative Paul D. Ryan of Wisconsin, the chairman of the House Budget Committee, said Republicans were now willing to go to formal negotiations with Senate Democrats over a long-term, comprehensive budget framework, a move Republicans have resisted since April. And while House Republicans are divided over even a short-term increase in the debt ceiling, Representative Tim Griffin, Republican of Arkansas, said the proposal would pass with Republican and Democratic votes.

Many Republicans said they would support their leadership’s proposal only if Mr. Obama agreed to not only negotiate a broader deal on a yearlong debt ceiling increase — which would most likely include spending and entitlement changes — but also negotiate over the terms for a stopgap spending measure that would reopen the government.

“We would get a commitment from the president, a concession, to sit down and talk with us, to negotiate before we go to a yearlong debt ceiling increase,” said Representative John Fleming, Republican of Louisiana. “There won’t be a vote on a temporary extension of the debt limit unless the president agrees this afternoon to sit down and talk with us.”

In a closed-door meeting Thursday morning, Republican lawmakers argued, alternately, for both a more moderate and a more hard-line approach. Representative Charlie Dent, a Republican who represents a swing district in Eastern Pennsylvania, stood up and pushed for a vote on a stopgap measure to finance the government with no strings attached, according to members who were in the room. But other more conservative members, Mr. Fleming said, are “concerned that we don’t move the goal posts unless we get the president to come to us, that this has to work bilaterally.”

Leaving the meeting early, Representative Steve King, Republican of Iowa and one of his party’s more conservative members, said he was “not very enthusiastic” about the new strategy.

Still, despite continuing divisions within the Republican ranks and between the parties, the maneuvering suggested that Congressional Republicans want to end a potentially calamitous standoff for which polls show they are getting more of the blame from the public.

“When you’re genuinely convinced your own constituency — Democrats or Republicans — are beginning to waver, then that’s when you start closing down and moving forward,” said Representative Jack Kingston, Republican of Georgia, who is running for the Senate.

He said that had not happened yet, “but I do think that what reopens shutdowns more than policy are polls.”

Economists across a broad spectrum agree that violating the debt limit — which could happen as early as next week, absent Congressional action — would probably severely damage the economy. The new Republican proposal could temporarily remove that threat.

Treasury Secretary Jacob J. Lew implored Congress on Thursday to raise the debt ceiling, warning the Senate Finance Committee of potentially severe market and economic repercussions if it did not.

In testimony before the committee, Mr. Lew stressed that the Treasury Department would run out of “extraordinary measures” to free up cash in a matter of days. At that point, the country’s bills might overwhelm its cash on hand in addition to any receipts from taxes or other sources, leading to an unprecedented default.

Mr. Lew said that the Treasury had no workarounds to avoid breaching the debt ceiling. “There is no plan other than raising the debt limit,” he said. “The legal issues, even regarding interest and principal on the debt, are complicated.”

In not addressing the shutdown, Republicans would keep pressure on Democrats to negotiate on a package of deficit reduction and tax overhaul proposals.

Mr. Griffin, the Arkansas Republican, said that the plan the speaker presented appealed to him because it would give House Republicans some breathing room as they continued to try to plot their next steps.

“We can’t even contemplate putting our full faith and credit, our credit rating and our financial system at risk, and this makes that clear,” he said.

House Republican leadership aides said the Nov. 22 date was picked to coincide with the Thanksgiving recess to add pressure on both parties to come to a compromise. But with so little time, Mr. Boehner conceded, “You could end up back in the same place.”

Representative Mark Meadows, a freshman Republican from North Carolina who was an initial champion of the “defund Obamacare” movement, said he was in favor of the new strategy.

“Really, what this is all about is negotiating and finding common ground,” he said. “This looks like a window to be able to do that, and I’m supportive.”

Meanwhile, a group of Republican senators began meeting with Senator Mitch McConnell of Kentucky, the Republican leader, to find a bipartisan solution to the twin fiscal impasses.

The senators are examining a yearlong resolution to reopen the government and finance it at levels that reflect the automatic spending cuts known as sequestration, but with added flexibility to help government agencies and departments deal with the tight budgets. The plan would also include a short-term increase in the debt ceiling, reflecting the House’s plan, and would include a repeal of a medical device tax unpopular with some Democrats.

The strategy is similar to a plan proposed earlier this week by Senator Susan Collins, Republican of Maine, but it would also tighten income verification rules for the new health care exchanges.

Senate Republicans have begun shopping this plan around to their Democratic counterparts, and they hope to attach it to the debt ceiling extension that the House sends over, before sending it back for a vote.

During the Senate hearing, Republicans chastised the administration for focusing on the debt limit more than the country’s debt, and for refusing to negotiate. Democrats and Republicans generally talked past one another at the hearing, with little agreement on how to raise the ceiling and how to manage the budget.

If the Obama administration will not negotiate over entitlement programs like Medicare now, “when will they negotiate?” asked Senator Orrin G. Hatch, Republican of Utah.

Mr. Lew said that the Obama administration would be willing to sit down and negotiate longer-term budgetary changes, but would not tie such negotiations to the debt ceiling. He would not say how long or how big a debt ceiling increase the administration wanted.

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